A GENERALIZED POLYGAMMA FUNCTION
OLIVIER ESPINOSA AND VICTOR H. MOLL
Abstract. We study the properties of a function ψ(z, q) (the generalized
polygamma function), defined for complex values of the variables z and q,
which is entire in the variable z and reduces to the usual polygamma function
ψ (m) (q) for z a non-negative integer m, and to the balanced negapolygamma
function ψ (−m) (q) introduced in [5] for z a negative integer −m.
1. Introduction
The Hurwitz zeta function defined by
(1.1)
ζ(z, q)
∞
X
1
(n + q)z
n=0
=
for z ∈ C, Re z > 1 and q 6= 0, −1, −2, . . . is a generalization of the Riemann zeta
function ζ(z) = ζ(z, 1). This function admits a meromorphic continuation into the
whole complex plane. The only singularity is a simple pole at z = 1 with unit
residue. The recent paper [9] presents a motivated discussion of this extension.
The Hurwitz zeta function turns out to be related to the classical gamma function, defined for Re q > 0 by
Z ∞
(1.2)
Γ(q) =
tq−1 e−t dt,
0
in several different ways. For example, the digamma function
(1.3)
ψ(q)
=
d
log Γ(q)
dq
appears in the Laurent expansion of ζ(z, q) at the pole z = 1:
(1.4)
ζ(z, q) =
1
− ψ(q) + O(z − 1).
z−1
A second connection among these functions is given by Lerch’s identity
Γ(q)
(1.5)
ζ 0 (0, q) = log Γ(q) + ζ 0 (0) = log √
2π
√
where we have used the classical value ζ 0 (0) = − log 2π in the last step.
Date: November 12, 2002.
1991 Mathematics Subject Classification. Primary 33.
Key words and phrases. Hurwitz zeta function, polygamma function, negapolygammas.
1
2
OLIVIER ESPINOSA AND VICTOR H. MOLL
A third example is the relation between the Hurwitz zeta function and the polygamma function defined by
dm
(1.6)
ψ (m) (q) = m ψ(q), m ∈ N,
dq
namely
(1.7)
ψ (m) (q)
= (−1)m+1 m! ζ(m + 1, q).
These relations are not independent. Both (1.5) and (1.7) can be derived from (1.4)
with the aid of the formula
∂
(1.8)
ζ(z, q) = −z ζ(z + 1, q).
∂q
The digamma (ψ(q) = ψ (0) (q)) and polygamma functions are analytic everywhere in the complex q-plane, except for poles (of order m + 1) at all non-positive
integers. The residues at these poles are all given by (−1)m+1 m!.
Extensions of the polygamma function ψ (m) (q) for m a negative integer have been
defined by several authors [1, 6, 5]. These functions have been called negapolygamma
functions. For example, Gosper [6] defined the family of functions
(1.9)
ψ−1 (q) := log Γ(q),
Z q
ψ−k (q) :=
ψ−k+1 (t)dt,
k ≥ 2,
0
which were later reconsidered by Adamchik [1] in the form
Z q
1
(1.10)
ψ−k (q) =
(q − t)k−2 log Γ(t)dt, k ≥ 2.
(k − 2)! 0
These negapolygamma functions can be expressed in terms of the derivative (with
respect to its first argument) of the Hurwitz zeta function at the negative integers
[1, 6]. The definition of the negapolygamma functions in (1.9) can be modified by
introducing arbitrary constants of integration at every step. This yields infinitely
many different families of negapolygamma functions, with the property that the
corresponding members of any two families differ by a polynomial,
(−m)
ψa(−m) (q) − ψb
(q) = pm−1 (q),
where the functions pn (q) are polynomials in q of degree n, satisfying the property
d
pn+1 (q).
dq
An example of such modified negapolygamma functions has been introduced in
[5], in connection with integrals involving the polygamma and the loggamma functions. These are the balanced negapolygamma functions, defined for m ∈ N by
pn (q) =
(1.11)
ψ (−m) (q) :=
1
[Am (q) − Hm−1 Bm (q)] ,
m!
Here Hr := 1 + 1/2 + · · · + 1/r is the harmonic number (H0 := 0), Bm (q) is the
m-th Bernoulli polynomial, and the functions Am (q) are defined in terms of the
Hurwitz zeta function as
(1.12)
Am (q) := m ζ 0 (1 − m, q).
GENERALIZED POLYGAMMA FUNCTION
3
A function f (q) is defined to be balanced (on the unit interval) if it satisfies the two
properties
Z 1
f (q)dq = 0 and f (0) = f (1).
0
Note that the Bernoulli polynomials, which are related to the Hurwitz zeta function
in a way similar to (1.12),
(1.13)
Bm (q) = −m ζ(1 − m, q),
m ∈ N,
are themselves balanced functions. In [5] we have shown that the balanced negapolygamma functions (1.11) satisfy
(1.14)
d (−m)
ψ
(q) = ψ (−m+1) (q),
dq
m ∈ N.
This makes them a negapolygamma family, connecting ψ (−1) (q) = log Γ(q) + ζ 0 (0)
to the digamma function ψ (0) (q) = d log Γ(q)/dq.
The goal of this work is to introduce and study a meromorphic function of two
complex variables, ψ(z, q), the generalized polygamma function, that reduces to the
polygamma function ψ (m) (q) for z = m ∈ N0 and to the balanced negapolygamma
function ψ (−m) (q) for z = −m ∈ −N. We describe some analytic properties of
ψ(z, q) and show they extend those of polygamma and balanced negapolygamma
functions. We also present some definite integral formulas involving ψ(z, q) in the
integrand. Finally, we compare our generalized polygamma function with a different generalization introduced by Grossman in [8].
2. The generalized polygamma function
The generalized polygamma function is defined by
∂
ζ(z + 1, q)
ψ(z, q) := e−γz
(2.1)
eγz
,
∂z
Γ(−z)
where z ∈ C and q ∈ C, q 6∈ −N0 . At z = m ∈ N, where Γ(−z) has a pole,
and at z = 0, where both Γ(−z) and ζ(z + 1, q) have poles, we define (2.1) by its
corresponding limiting values. We show below that, for fixed q, ψ(z, q) is indeed an
entire function of z.
The alternative representation
eγz ∂ζ(z, q)
−γz ∂
(2.2)
ψ(z, q) = e
∂z Γ(1 − z) ∂q
follows directly from (1.8).
Lemma 2.1. The function ψ(z, q) is given by
i
1 h 0
(2.3)
ψ(z, q) =
ζ (z + 1, q) + {γ + ψ(−z)} ζ(z + 1, q)
Γ(−z)
and
(2.4)
ψ(z, q) =
i
1 h 0
ζ (z + 1, q) + H(−z − 1)ζ(z + 1, q) ,
Γ(−z)
4
OLIVIER ESPINOSA AND VICTOR H. MOLL
where H is defined by
(2.5)
H(z) :=
∞ X
1
k=1
k
−
1
k+z
.
Proof. Differentiation of (2.1) yields (2.3). The second representation follows from
the identity H(z) = γ + ψ(z + 1); see [12], page ??, for details.
The function H is called the generalized harmonic number function. It has simple
poles with residue −1 at all negative integers, and reduces to the n-th harmonic
number Hk for z = k ∈ N0 . H(z) satisfies the following reflection formula:
H(−z) = H(z − 1) + π cot πz.
(2.6)
We show first that, for m ∈ N, ψ(−m, q) reduces to the balanced negapolygamma
function ψ (−m) (q) defined in (1.11).
Theorem 2.2. For m ∈ N, ψ(−m, q) = ψ (−m) (q).
Proof. Lemma 2.1 gives
(2.7)
ψ(−m, q)
=
1
[ζ 0 (1 − m, q) + Hm−1 ζ(1 − m, q)] .
Γ(m)
The result now follows from (1.11), (1.12), and (1.13).
We show next that the generalized polygamma function has no singularities in
the complex z plane and that ψ(0, q) is actually the digamma function ψ(q).
Theorem 2.3. For fixed q ∈ C, the function ψ(z, q) is an entire function of z.
Moreover ψ(0, q) = ψ(q).
Proof. In the representation (2.4), the term 1/Γ(z) is entire and ζ(z, q) has only
a simple pole at z = 1 and is analytic for z 6= 1. Thus z = 0 is the only possible
singularity for ψ(z, q). This singularity is removable because for z near 0
ζ 0 (z + 1, q)
1
= − γ + O(z)
Γ(−z)
z
and
H(−z − 1) ζ(z + 1, q)
1
= − + γ + ψ(q) + O(z),
Γ(−z)
z
so that ψ(z, q) = ψ(q) + O(z).
We finally show that, for m ∈ N, ψ(m, q) reduces to the polygamma function
ψ (m) (q) defined in (1.6).
Theorem 2.4. The function ψ(z, q) satisfies
(2.8)
∂
ψ(z, q) = ψ(z + 1, q)
∂q
and
(2.9)
ψ(m, q) = ψ (m) (q),
m ∈ N.
GENERALIZED POLYGAMMA FUNCTION
5
Proof. Use (1.8) to produce
∂
−γz ∂
γz (z + 1) ζ(z + 2, q)
ψ(z, q) = −e
e
∂q
∂z
Γ(−z)
and then use Γ(−z) = −(z + 1)Γ(−z − 1) to obtain (2.8).
The identity (2.9) follows by induction from Theorem 2.3 and (2.8), but we
provide an alternative proof. Set z = m + and consider (2.4) as → 0. The
expansions
1
1
= (−1)m+1 m! + O(2 ) and H(−1 − m − ) = + Hm + O()
Γ(−m − )
are the only terms that produce a nonvanishing contribution in (2.4) as → 0.
We conclude that ψ(m, q) = (−1)m+1 m! ζ(m + 1, q) and the result follows from
(1.7).
3. Functional relations
The generalized polygamma function ψ(z, q), as a function of q, satisfies some
simple algebraic and analytic relations. These are derived from those of Γ(z) and
ζ(z, q).
Theorem 3.1. The function ψ(z, q) satisfies
(3.1)
ψ(z, q + 1) = ψ(z, q) +
ln q − H(−z − 1)
.
q z+1 Γ(−z)
Proof. The identity
(3.2)
ζ(z, q + 1) = ζ(z, q) −
1
,
qz
produces
∂
1
γz
ψ(z, q + 1) = ψ(z, q) − e
e
.
∂z
q z+1 Γ(−z)
The result now follows from γ + ψ(−z) = H(−z − 1).
−γz
Relation (3.1) generalizes the well-known functional relations for the digamma
and polygamma functions,
1
ψ(q + 1) = ψ(q) + ,
(3.3)
q
(−1)m m!
(3.4)
ψ (m) (q + 1) = ψ (m) (q) +
,
q m+1
and the corresponding relation
(3.5)
ψ (−m) (q + 1)
= ψ (−m) (q) +
q m−1
[ln q − Hm−1 ]
(m − 1)!
for the balanced negapolygamma function [5].
Note. We have been unable to find a generalization of the other well-known functional relation for the polygamma function,
dm
(3.6)
(−1)m ψ (m) (1 − q) = ψ (m) (q) + m π cot πq.
dq
6
OLIVIER ESPINOSA AND VICTOR H. MOLL
The next result establishes a multiplication formula for ψ(z, q). It generalizes
the analogous result for the digamma function, [7] (8.365.6).
Theorem 3.2. Let k ∈ N. Then,
(3.7)
k z+1 ψ(z, kq) =
k−1
X
ψ (z, q + j/k) − k z+1 ln k
j=0
=
k−1
X
ψ (z, q + j/k) −
j=0
ζ(z + 1, kq)
Γ(−z)
ln k
ζ (z + 1, q + j/k) .
Γ(−z)
Proof. Use the multiplication rule
k z ζ(z, kq) =
(3.8)
k−1
X
ζ (z, q + j/k)
j=0
for the Hurwitz zeta function in the definition (2.1) of ψ(z, q).
The case k = 2 yields the duplication formula
(3.9)
ψ(z, 2q) =
1
ζ(z + 1, 2q)
[ψ(z, q) + ψ(z, q + 1/2)] − ln 2
.
2z+1
Γ(−z)
4. Series expansions of ψ(z, q)
In this section we present two different series expansions for the generalized
polygamma function. The first is a generalization of the well-known expansion of
the digamma function,
(4.1)
ψ(q + 1) = −γ +
∞
X
(−1)k+1 ζ(k + 1)q k ,
|q| < 1.
k=1
Theorem 4.1. Let z ∈ C and |q| < 1. Then
(4.2)
ψ(z, q + 1) =
∞
X
ψ(z + k, 1)
k=0
qk
.
k!
Proof. The Taylor expansion of ψ(z, q + 1) around q = 0 can be expressed in terms
of ψ(z, q) using the iterated version of (2.8),
∂k
ψ(z, q) = ψ(z + k, q)
∂q k
(4.3)
evaluated at q = 1. The radius of convergence is computed to be 1 by using the
ratio test, the identity
(4.4)
ψ(z, 1) =
1
[ζ 0 (z + 1) + H(−z − 1) ζ(z + 1)] ,
Γ(−z)
and the fact that ζ 0 (z + 1) tends to zero faster than the term H(−z − 1) ζ(z + 1)
as z → ∞.
GENERALIZED POLYGAMMA FUNCTION
7
Note 4.2. The series expansion (4.1) for the digamma function is the special case
z = 0 of (4.2). This follows from the values
ψ(0, 1) = ψ(1) = −γ,
and
ψ(k, 1) = ψ (k) (1) = (−1)k+1 k! ζ(k + 1)
for k ∈ N. Similarly z = −1 and the value ψ(−1, 1) = ζ 0 (0) yield the well-known
expansion of the loggamma function,
(4.5)
log Γ(q + 1)
= −γq +
∞
X
k=2
(−1)k
ζ(k) k
q ,
k
|q| < 1.
Note 4.3. Riemann’s functional equation,
(4.6)
ζ(z) (2π)1−z
2Γ(1 − z) sin(πz/2)
πz ζ(z)Γ(z)
= 2 cos
,
2
(2π)z
ζ(1 − z) =
yields the alternate representation
πz h
i
π
πz (4.7)
ψ(z, 1) = 2 (2π)z cos
γ + ln 2π − tan
ζ(−z) − ζ 0 (−z) .
2
2
2
Note 4.4. Theorems 3.1 and 4.1 determine the behavior of ψ(z, q) for small q:
(4.8)
ψ(z, q) = −
ln q
H(−z − 1) 1
1
+
+ ψ(z, 1) + ψ(z + 1, 1)q + O(q 2 ).
z+1
Γ(−z) q
Γ(−z) q z+1
For z = m ∈ N0 the coefficients of the first two terms are
1
=0
Γ(−m)
and
H(−m − 1)
= (−1)m+1 m!,
Γ(−m)
so the logarithmic term drops out and we recover the known behavior of the
polygamma function as q → 0,
(4.9)
ψ (m) (q) =
(−1)m+1 m!
+ ψ (m) (1) + O(q).
q m+1
For z 6∈ N with Re z ≥ −1 the first term in (4.8) determines the leading behavior,
and if Re z < −1 the first two terms in (4.8) vanish as q → 0 and hence ψ(z, q)
tends to the finite value ψ(z, 1) given by (4.4) or (4.7).
We now establish a Fourier series representation for the generalized polygamma
function ψ(z, q).
Theorem 4.5. For Re z < −1 and 0 ≤ q ≤ 1:
"∞
X
z
(4.10) ψ(z, q) = 2(2π)
nz (γ + ln 2πn) cos(2πnq + πz/2)
n=1
#
∞
πX z
−
n sin(2πnq + πz/2) .
2 n=1
8
OLIVIER ESPINOSA AND VICTOR H. MOLL
This result generalizes the Fourier expansion for the balanced negapolygamma function given in [5]. It implies that ψ(z, q) is itself balanced for any z such that
Re z < −1.
Proof. Let s = z + 1 in the Fourier representation of the Hurwitz zeta function,
"
#
∞
∞
πs X
πs X
cos(2πqn)
sin(2πqn)
2Γ(1 − s)
(4.11) ζ(s, q) =
sin
+ cos
,
(2π)1−s
2 n=1 n1−s
2 n=1 n1−s
which is valid for Re s < 0 and 0 ≤ q ≤ 1, and substitute (4.11) into (2.1).
5. Integral representations of ψ(z, q)
This section contains integral representations for ψ(z, q) that are derived directly
from corresponding integral representations of the Hurwitz zeta function. For instance,
Z ∞ −qt
1
e
(5.1)
ζ(z, q) =
tz−1 dt,
Γ(z) 0 1 − e−t
valid for Re z > 1 and Re q > 0, implies the next result.
Theorem 5.1. Let Re z > 0 and Re q > 0. Then
Z ∞ −qt z e t
γ
sin πz
(5.2)
ψ(z, q) = −
cos
πz
+
sin
πz
+
ln
t
dt.
1 − e−t
π
π
0
Proof. The identity
(5.3)
ζ(z + 1, q)
sin πz
=−
Γ(−z)
π
Z
∞
0
e−qt z
t dt
1 − e−t
follows from the integral representation for ζ(z, q) in (5.1) and the reflection rule
for the gamma function. The result now follows from the definition of ψ(z, q). A Hankel-type contour is a curve that starts at +∞ + i 0+, moves to the left on
the upper half-plane, encircles the origin once in the positive direction, and returns
to +∞ − i 0+ on the lower half-plane. The Hurwitz zeta function has the following
integral representation along a Hankel-type contour[12]:
Z (0+) −qt
ζ(z + 1, q)
1
e
(5.4)
=−
(−t)z dt,
Γ(−z)
2πi ∞
1 − e−t
valid for arbitrary complex z and Re q > 0.
Theorem 5.2. Let q, z ∈ C with Re q > 0. Then
Z (0+)
1
[γ + ln(−t)]e−qt
(5.5)
ψ(z, q) = −
(−t)z dt.
2πi ∞
1 − e−t
Proof. The result follows directly from (5.4).
GENERALIZED POLYGAMMA FUNCTION
9
6. Definite integrals involving ψ(z, q)
Definite integrals of ψ(z, a + bq) can be directly obtained from its primitive,
Z
(6.1)
ψ(z, a + bq) dq = b−1 ψ(z − 1, a + bq),
according to (2.8). So, for example,
(
Z 1
0,
(6.2)
ψ(z, q) dq =
∞,
0
if Re z < 0,
if Re z ≥ 0,
where we have used the result (4.8) to evaluate ψ(z, q) at the origin.
The integral formulas presented below are direct consequences of the corresponding integral formulas for the Hurwitz zeta function. Several of these were derived
in [4, 5].
Theorem 6.1. For Re z, Re z 0 < 0 and Re(z + z 0 ) < −1,
(6.3)
Z 1
0
z+z 0
ψ(z, q)ψ(z , q) dq = 2(2π)
0
π(z − z 0 )
cos
2
(
π2
2
+ (γ + ln 2π) ζ(−z − z 0 )
4
)
− 2 (γ + ln 2π) ζ 0 (−z − z 0 ) + ζ 00 (−z − z 0 ) .
Proof. This is a direct consequence of the following result [4],
Z 1
π(s − s0 )
2Γ(1 − s)Γ(1 − s0 )
ζ(2 − s − s0 ) cos
,
ζ(s, q)ζ(s0 , q)dq =
0
2−s−s
(2π)
2
0
valid for Re s < 1, Re s0 < 1, Re(s + s0 ) < 1. Set s = z + 1, s0 = z 0 + 1, divide by
Γ(−z)Γ(−z 0 ), and construct the functions ψ(z, q), ψ(z 0 , q) in the integrand according to definition (2.1).
The evaluation (6.3) generalizes Example 5.6 in [5]: for k, k 0 ∈ N,
"
Z 1
2 cos(k − k 0 ) π2 00
(−k)
(−k0 )
ψ
(q)ψ
(q) dq =
ζ (k + k 0 ) − 2(γ + ln 2π)ζ 0 (k + k 0 )
k+k0
(2π)
0
#
π2
2
0
+ (γ + ln 2π) +
ζ(k + k ) .
4
The special case k = k 0 = 1 reduces to
Z 1
√
√
γ2
π2
1
4
2
(ln Γ(q)) dq =
+
+ γ ln 2π + ln2 2π
12
48 3
3
0
0
√
ζ (2) ζ 00 (2)
− (γ + 2 ln 2π) 2 +
,
π
2π 2
given in [4].
10
OLIVIER ESPINOSA AND VICTOR H. MOLL
Corollary 6.2. Let Re z < −1/2. Then
(
Z 1
π2
2
2
2z
(6.4)
ψ(z, q) dq = 2(2π)
+ (γ + ln 2π) ζ(−2z)
4
0
0
00
)
− 2 (γ + ln 2π) ζ (−2z) + ζ (−2z) .
For Re z < −1,
Z
(6.5)
1
ψ(z, q)ψ(z + 1, q) dq = 0.
0
The same type of argument gives the next evaluation.
Theorem 6.3. For Re z, Re z 0 < 0, and Re(z + z 0 ) < −1,
(
Z 1
0
π
π
(6.6)
ζ(z + 1, q)ψ(z 0 , q)dq = 2(2π)z+z Γ(−z)
ζ(−z − z 0 ) sin (z − z 0 )
2
2
0
)
i
π
0
+ (γ + ln 2π)ζ(−z − z ) − ζ (−z − z ) cos (z − z ) .
2
h
0
0
0
Corollary 6.4. For Re z < 0,
Z 1
1
(6.7)
ζ(z, q)ψ(z, q)dq = − (2π)2z Γ(1 − z)ζ(1 − 2z).
2
0
Our final evaluation computes the Mellin transform of the generalized polygamma
function.
Theorem 6.5. Let a, b ∈ R+ , α, z ∈ C such that 0 < Re α < Re z. Then
Z ∞
b−α Γ(α) h
(sin πz)ψ(z − α, a)
(6.8)
q α−1 ψ(z, a + bq) dq =
sin π(z − α)
0
i
+ (sin πα)Γ(z + 1 − α)ζ(z + 1 − α, a) .
Proof. Start from formula (2.3.1.1) of [10],
Z ∞
q α−1 ζ(s, a + bq)dq = b−α B(α, s − α)ζ(s − α, a),
0
valid for a, b ∈ R+ and 0 < Re(α) < Re(s) − 1, set s = z + 1, and use the definition
(2.1) of ψ(z, q) to evaluate the integral as
b−α Γ(α) −γz ∂ γz
e
[e (sin πz)Γ(z + 1 − α)ζ(z + 1 − α, a)] .
π
∂z
The desired evaluation now follows from the reflection formulas for the gamma and
digamma functions,
π
Γ(1 − x)Γ(x) =
and ψ(1 − x) = ψ(x) + π cot πx,
sin πx
respectively.
−
GENERALIZED POLYGAMMA FUNCTION
11
Note 6.6. The special case z = m ∈ N in Theorem 6.5 yields an explicit form for
the Mellin transform of the polygamma function ψ (m) (a + bq):
Z ∞
(6.9)
q α−1 ψ (m) (a + bq) dq = (−1)m+1 b−α Γ(α)Γ(1 + m − α)ζ(1 + m − α, a),
0
valid when 0 < Re α < m. This formula generalizes formula (6.473) of [7] to the
case a, b 6= 1.
7. Relation to Grossman’s generalization of the polygamma function
In 1975, N. Grossman presented a generalization of polygamma functions to
arbitrary complex orders [8] which is different to ours. He was motivated by a
problem proposed a year earlier by B. Ross [11] concerning the convergence and
evaluation of the integral
Z q
(7.1)
I=
(q − t)p−1 log Γ(t)dt.
0
For p ∈ N, this integral corresponds precisely (up to a normalization factor) to the
Gosper-Adamchik’s negapolygamma functions defined by (1.10). In [8] the author
used the techniques of Liouville’s fractional integration and differentiation to obtain
a generalization ψ (ν) (q) of the polygamma function, with ν ∈ C, in the form
q −ν−1
1
Γ0 (−ν)
γq −ν
(7.2) ψ (ν) (q) =
ln + γ +
−
Γ(−ν)
q
Γ(−ν)
Γ(1 − ν)
−ν−1 Z λ+i∞
q
Γ(z)ζ(z) π
qz
dz,
−
2πi λ−i∞
Γ(z − ν) sin πz
where the contour of integration is along a vertical line with 1 < λ < 2. The function ψ (ν) (q) is an entire function in the ν-plane, for each q in the plane cut along
the negative real axis [8].
For ν = −m ∈ −N0 , Grossman’s generalized polygamma ψ (ν) (q) reduces to the
Gosper-Adamchik negapolygamma functions ψ−m (q). We showed in [5] that the
latter are related to the balanced negapolygammas ψ (−m) (q) by
(7.3)
ψ
(−m)
(q) = ψ−m (q) +
m−1
X
r=0
q m−r−1
[ζ 0 (−r) + Hr ζ(−r)] ,
r!(m − r − 1)!
which, in light of (4.4), can be also expressed as
(7.4)
ψ (−m) (q) = ψ−m (q) +
m−1
X
r=0
q m−r−1
ψ(−r − 1, 1).
Γ(m − r)
The relation between the functions ψ(ν, q) and ψ (ν) (q) needs to be explored. Since
both of these functions are entire in the complex variable ν, their difference
(7.5)
Ψ(ν, q) := ψ(ν, q) − ψ (ν) (q)
must be an entire function itself. Furthermore, since both ψ(ν, q) and ψ (ν) (q) reduce
to the standard polygamma function when ν ∈ N0 , Ψ(ν, q) vanishes at ν ∈ N0 . In
12
OLIVIER ESPINOSA AND VICTOR H. MOLL
order to study further properties of the function Ψ(ν, q), we shall consider the
asymptotic and small-q series expansions of both ψ(ν, q) and ψ (ν) (q). Let
Z λ+i∞
Γ(z)ζ(z) π
1
(7.6)
I(ν, q) =
qz
dz.
2πi λ−i∞
Γ(z − ν) sin πz
For q > 1, Grossman suggested to deform the contour so that it starts at −∞−i0+,
runs below the real axis, encircles the point z = 1 in the positive sense (crossing the
real axis to the left of z = 2), and then returns to −∞ + i0+ running over the real
axis. I(ν, q) can be evaluated along the deformed contour by a residue calculation.
The only relevant poles are z = 1 and z = 0, −1, −2, . . ., coming from Γ(z), ζ(z),
and from the zeros of sin(πz). The poles at z = −2k are simple since ζ(−2k) = 0.
All the other poles are double. Let Rν (z0 ) be the residue at the pole z = z0 . Then
−q ln q + qψ(1 − ν)
,
Γ(1 − ν)
H(−1 − ν) − ln 2π − ln q
Rν (0) =
,
2Γ(−ν)
Bm+1
1
0
ζ
(−m)
−
{ln
q
+
H
−
H(−m
−
ν
−
1)}
,
Rν (−m) =
m
m!Γ(−m − ν)q m
m+1
Rν (1) =
for m = 1, 2, 3, . . .. Using the special values B0 = 1, B1 = −1/2, ζ 0 (0) = − 12 ln 2π,
and H0 = 0, we obtain
(
∞
∞
X
X
kζ 0 (1 − k) − Bk Hk−1
Bk
(ν)
−ν
(7.7) ψ (q) = q
−
ln q
k
k!Γ(1 − ν − k)q
k!Γ(1 − ν − k)q k
k=0
k=1
)
∞
X
Bk H(−k − ν)
−
.
k!Γ(1 − ν − k)q k
k=0
We observe that Grossman [8] missed the logarithmic contribution.
The asymptotic expansion of the generalized polygamma function ψ(ν, q) for
large q can also be obtained from (2.1) and the asymptotic expansion of ζ(z, q)
itself. This yields
(7.8)
ψ(ν, q) ∼ q
−ν
(
ln q
∞
∞
X
sin πν X
Bk Γ(k + ν)
Bk Γ(k + ν)
(−1)k
−
cos
πν
(−1)k
k
π
k!
q
k!
qk
k=0
k=0
)
∞
sin πν X
B
H(k
+
ν
−
1)Γ(k
+
ν)
k
(−1)k
.
−
π
k!
qk
k=0
The reflection formula for Γ(z) yields
sin πν
1
(−1)k Γ(k + ν) =
,
π
Γ(1 − ν − k)
and the reflection formula (2.6) for the harmonic number function produces
H(k + ν − 1) = H(−k − ν) − π cot πν.
GENERALIZED POLYGAMMA FUNCTION
13
Thus
(7.9)
ψ(ν, q) ∼ ln q
∞
X
k=0
∞
X Bk H(−ν − k)
Bk
−
.
k!Γ(1 − ν − k)q k+ν
k!Γ(1 − ν − k)q k+ν
k=0
We obtain therefore the following asymptotic expansion for the function Ψ(ν, q)
defined by (7.5):
Ψ(ν, q) ∼
(7.10)
∞
X
k=1
ψ(−k, 1)
.
Γ(1 − ν − k)q k+ν
We note that for ν = m ∈ N0 the formal series on the right-hand side vanishes
identically, as it should. For ν = −m ∈ −N, the series above reduces to a polynomial in q, which coincides with the one appearing in (7.4).
On the other hand, for |q| < 1, Grossman has proven that his polygamma function has the convergent expansion
(
)
∞
γq X
q −ν−1
(ν)
k
k
− ln q + γ + ψ(−ν) +
+
(−1) ζ(k)B(−ν, k)q
,
(7.11) ψ (q) =
Γ(−ν)
ν
k=2
which, on account of the special values for ψ(z, 1) at the non-negative integers given
in (4.7), can be written as
∞
(7.12)
ψ (ν) (q) =
− ln q + H(−ν − 1) X
ψ(k, 1)
+
q k−ν .
ν+1
q
Γ(−ν)
Γ(−ν + k + 1)
k=0
This is to be compared with the small-q expansion of the generalized polygamma
function ψ(ν, q) obtained in Theorems 3.1 and 4.1:
∞
(7.13)
ψ(ν, q) =
− ln q + H(−ν − 1) X ψ(k + ν, 1) k
+
q .
q ν+1 Γ(−ν)
Γ(k + 1)
k=0
Again, both expansions coincide if ν ∈ N0 , since 1/Γ(z) vanishes at the non-positive
integers, and differ by the polynomial in (7.4) if ν = −m ∈ −N.
Acknowledgments. The first author would like to thank the Department of Mathematics at Tulane University for its hospitality. The second author acknowledges
the partial support of NSF-DMS 0070567.
References
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1998, 191-199.
[2] BOROS, G. - ESPINOSA, O. - MOLL, V.: On some families of integrals solvable in terms
of polygamma and negapolygamma functions. Preprint.
[3] BROMWICH, T.J.: An Introduction to the Theory of Infinite Series, 2nd. Edition, MacMillan, New York, 1926.
[4] ESPINOSA, O. - MOLL, V.: On some integrals involving the Hurwitz zeta function: part 1.
The Ramanujan Journal, 6, 2002, 159-188.
[5] ESPINOSA, O. - MOLL, V.: On some integrals involving the Hurwitz zeta function: part 2.
The Ramanujan Journal, 6, 2002, 449-468.
R m/6
[6] GOSPER, R. Wm. Jr.: n/4 ln Γ(z)dz. In Special functions, q-series and related topics, pages
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AMS, 1997.
14
OLIVIER ESPINOSA AND VICTOR H. MOLL
[7] GRADSHTEYN, I.S. - RYZHIK, I.M.: Table of Integrals, Series and Products. Fifth edition,
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[8] GROSSMAN, N.: Polygamma functions of arbitrary order. SIAM J. Math. Anal. 7, 1976,
366-372.
[9] KNOPP, M. - ROBINS, S.: Easy proofs of Riemann’s functional equation for ζ(s) and of
Lipschitz summation. Proc. AMS 129, 2001, 1915-1922.
[10] PRUDNIKOV, A.P.- BRYCHKOV, Yu. A. - MARICHEV, O.I.: Integrals and Series. Volume
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Science Publishers, New York, 1990.
[11] ROSS, B.: Problem 6002. Amer. Math. Monthly 81, 1974, 1121.
[12] WHITTAKER, E. - WATSON, G.: A Course of Modern Analysis. Cambridge University
Press, Fourth Edition reprinted, 1963.
Departamento de Fı́sica, Universidad Téc. Federico Santa Marı́a, Valparaı́so, Chile
E-mail address: [email protected]
Department of Mathematics, Tulane University, New Orleans, LA 70118
E-mail address: [email protected]